Recall that if \(R\)  is a commutative ring, then there is an \(E_{\infty }\)  -ring spectrum \(HR\)  which represents cohomology with coeffcients in \(R.\)  Consider \(R = \mathbb {Z}_2,\)  the prime field of two elements, then as well known, \(H^{*}(H\mathbb {Z}_2, \mathbb {Z}_2) = \mathcal {A}_2\)  is the Steenrod algebra of \(\mathbb {Z}_2\) -cohomology operations. This is the graded homotopy of the \(H\mathbb {Z}_2\)  -module Map(\(H\mathbb {Z}_2 ,H\mathbb {Z}_2\) ), with (noncommutative) multiplication coming from a composition of cohomology operations. The algebra \(\mathcal {A}_2\)  is known to be a Hopf algebra (i.e., it is a bialgebra over the filed \(\mathbb {Z}_2\)  that has an associative multiplication and a coassociative comultiplication and is equipped with a counit, a unit and an antipode). As an algebra over \(\mathbb {Z}_2,\)  \(\mathcal {A}_2\)  is generated by the Steenrod squaring operations \(Sq^{i}\)  of degree \(i\)  for \(i\ge 0.\)  These squares act on \(H^{*}(X, \mathbb {Z}_2)\)  for any spectrum \(X.\)  Let \(X= \mathbb {R}P(\infty )\)  be the infinite real projective space. Then, it has been shown that \(H^{*}(\mathbb {R}P(\infty ), \mathbb {Z}_2)\cong \mathbb {Z}_2[x]\)  and that the (left) action of \(\mathcal {A}_2\)  on \(\mathbb {Z}_2[x]\)  can be described by the rule \(Sq^{i}(x^{a}) = \binom{a}{i}x^{a+i}.\)  This action is deduced by induction from the case \(a = 1\)  by using the comultiplication on \(\mathcal {A}_2.\)  Also based on the comultiplication, one can explicitly depicte the action of \(\mathcal {A}_2\)  on \(H^{*}((\mathbb {R}P(\infty ))^{\times t}, \mathbb {Z}_2)\cong P_t:= \mathbb {Z}_2[x_1, \ldots , x_t]\). So, as well known, \(P_t\)  is an unstable left \(\mathcal {A}_2\) -module. It should be noted that \(\mathcal {A}_2\)  is not an unstable module over itself in general since \(Sq^{1}Sq^{0} = Sq^{1}\ne 0.\)  The \(\mathbb {Z}_2\) -algebra \(P_t\)  is a connected \(\mathbb {Z}\) -graded algebra. That is, \(P_t = \mathbb {Z}_2\bigoplus (\bigoplus _{n > 0}(P_t)_n),\)  where \((P_t)_n\)  is the vector space of homogeneous polynomials of degree \(n\) . It is natural to ask: What is a minimal set of generators for \(P_t\)  as an \(\mathcal {A}_2\) -module? When we fix the degrees \(n\)  and the number of variables \(t\) , answering that question is the same as the problem of elementary linear algebra of finding the dimension of the quotient space, \((QP_t)_n := (\mathbb {Z}_2\otimes _{\mathcal {A}_2} P_t)_n \cong (P_t)_n/\sum _{i > 0}{\rm Im}(Sq^{i})\). Here \(\sum _{i > 0}{\rm Im}(Sq^{i}) = (P_t)_n\cap {\rm Ker}(\epsilon )P_t,\)  in which \(\epsilon \)  denotes the counit \(\mathcal {A}_2\rightarrow \mathbb {Z}_2,\)  and the Steenrod squaring operations \(Sq^{i}: (P_t)_{n-i}\rightarrow (P_t)_n\)  are defined, and their basic properties, such as Cartan's formula, are established. People have also shown that \((QP_t)_n\)  has a module structure over the group algebra \(\mathbb {Z}_2GL_t\)  of the usual general linear group \(GL_t\) . The above problem, which has first appeared in the work of Peterson [1]}, is usually called the hit problem for the Steenrod algebra. It is currently famously difficult and is of great interest to many authors. (See Kameko [2]}, Mothebe-Uys [3]}, the present author [4]}, [5]}, [6]}, Singer [7]}, Sum [8]}, [9]}, Wood [10]}, and others.)
